Integration by Parts - Choice of variables

[takarin]

Homework Statement

I'm getting different results when choosing my u & dv for Integration by Parts on the following integral:

$$\int 2x^3 e^x^2 dx$$
(Note, the exponent on 'e' is x^2)

This yields the correct solution:
u = $$x^2$$
dv = $$2x e^x^2 dx$$

du = $$2xdx$$
v = $$e^x^2$$

However, I have tried using this instead (*)

u = $$2x^3$$
dv = $$e^x^2 dx$$

du = $$6x^2 dx$$
v = $$(e^x^2) / 2x$$

and this is yielding the incorrect solution (see 3.)

Homework Equations

Integration by Parts:
$$\int udv = uv - \int vdu$$

The Attempt at a Solution

The correct solution turns out to be
$$x^2 e^x^2 - e^x^2 + C$$

When I use my other choice of variables (*), I get (using IBP)
$$\int 2x^3 e^x^2 dx = 2x^3 e^x^2 / 2x - \int e^x^2 / 2x * 6x^2 dx$$
which leads to:
$$x^2 e^x^2 - 3/2 e^x^2 + C$$

which is different from the other choice of variables.

I've looked over both choices of variables, and I don't know why the second choice (*) comes up with a different solution.

Thanks for the help!

 

[vela]

Your error is that the integral of $$e^{x^2}$$ isn't $$e^{x^2}/(2x)$$. (I'm assuming you meant to have parentheses on the bottom, but perhaps not. Regardless, the answer is incorrect either way.)

 

[takarin]

Ah! Silly me, I've been staring at it and completely overlooked that. Thanks for your quick reply, I'll avoid such carelessness in the future :)